This entry describes classes of examples of A-∞ category-valued FQFTs defined on a version of the symplectic category.
Let $(X.\omega)$ be a compact symplectic manifold. At least in good cases to this is associated a Fukaya category $Fuk(X)$ of Lagrangian submanifolds and an enlarged version $Fuk^#(X)$.
Write $X^-$ for the symplectiv manifold $(X,-\omega)$.
Now if $(X_j, \omega_j)$ for $j = 0,1$ are two Lagrangian submanifolds and $L_{01} \subset X^-_0 \times X_1$ a Lagrangian correspondence then we get an A-∞ functor $\phi(L_{01}) : Fuk^#(X_0) \to Fuk^#(X_1)$
(Wehrheim, Woodward)
For $L_{01} \subset X^{-}_0 \times X_1$ and $L_{12} \subset X^{-}_1 \times X_2$ Lagrangian submanifolds, assuming monotonicity and Maslov conditions we have an $A_\infty$-homotopy
where on the right we have a natural notion of composition of Lagrangian submanifolds.
(Wehrheim-Woodward 07, section 3.2)
This is the symplectic version of Mukai functors?.
For $X_0$ and $X_1$ a compact Riemann surfaces and $M(X_0), M(X_1)$
their moduli spaces of fixed determinant rank $n$-bundles, and for $Y_{01}$ a cobordism (compact, oriented) from $X_0$ to $X_1$ then consider
If $Y_{01}$ is elementary in that there exists a Morse function $Y \to \mathbb{R}$ with $\leq 1$ critical points then $L(Y_{01})$ is a Lagrangian correspondence.
The assignment
defines a 2+1-dimensional FQFT for connected cobordisms with values in A-∞ categories.
This is supposed to be the 2+1-dimensional part of Donaldson theory.
Other theories that fit into this framework:
symplectic Khovanov theory? (Seidel-Smith and Rezazodegab)
Heegard-Floer theory?
$X$ a surface $\mapsto$ $Fuk^# sym X$
elementary cobordisms $\mapsto$ $\Phi($vanishing cycle$)$
Write $X^-j = (X_j , -\omega_j)$ for a symplectic manifold with its symplectic form reversed.
For $(X_j, \omega_j)$ two symplectic manifolds, a Lagrangian correspondence is a Lagrangian submanifold of $X^-_0 \times X_1$, that is
with $dim(L_{0,1}) = \frac{1}{2}(dim(x_0) + dim(X_1))$
and
where $\pi_i$ are the two projections out of the product.
The composition of two Lagrangian submanifolds is
which is a Lagrangian correspondence in $X^-_0 \times X_2$ if everything is suitably smoothly embedded by $\pi_{02}$.
For $\phi : X_0 \to X_1$ a symplectomorphism we have
$graph(\phi) \subset X_0^- \times X_1$ is a Lagrangian correspondence and composition of syplectomorphisms corresponds to composition of Lagrangian correspondences.
Let $X$ be a manifold, $G= U(n)$ the unitary group, $P \to X$ a $G$-principal bundle and $D \to X$ a $U(1)$-bundle with connection.
Then there is the moduli space $M(X) = M(P,D)$ of connections on $P$ with central curvature and given determinant.
For example if $X$ has genus $g$ then
such that $\prod_{j=1}^g A_j B_j A_j^{-1} B_j^{-1} = diag(e^{2\pi i d/})/G$
Let $Y_{01}$ be a cobordism from $X_0$ to $X_1$ with extension
is a Lagrangian correspondence if $Y_{01}$ is sufficiently simple. Further assuming this we have for composition that